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Integrate e^-at(Si(t)-pi/2) from zero to infinity

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Evaluate the integral with boundaries 0 to infinity:
J(a) = e^-at(Si(t)-pi/2)dt.

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Solution Summary

We explain in detail how the integral can be computed.

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The integral:

J(a) = Integral from 0 to infinity of exp(-at)[Si(t) -pi/2] dt

can be evaluated as follows. Inserting the expression Si(t) = integral from 0 to t of sin(x)/x dx gives:

J(a) = Integral over t from 0 to infinity of exp(-at)[Integral from 0 to t of sin(x)/x dx -pi/2] dt

Since Integral from 0 to infinity of sin(x)/x dx = pi/2, we can write this as:

J(a) =- Integral over t from 0 to infinity of exp(-at) Integral from x = t to infinity of sin(x)/x dx dt =

=- Integral over t ...

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